The omni-transform in the renewal model and in single-channel queues

Abstract

Some basic results of the renewal model are effectively summarized by $$E\psi '(r) = E[\psi (x) - \psi (0)]/Ex,$$ (1) wherex is the random variableservice time, r is its associatedresidual time, Ψ ( ) is an arbitrary “well behaved” function, andE is the expectation operator. The process “waiting time for service of a new arrival”, denoted byw, is effectively summarized in the modelM/G/1 by $$E\psi (w) = (1 - \rho )\psi (0) + \rho E\psi (w + r).$$ (2) We refer toEΨ (Z) as theomni-transform of the random variable or processZ, and to equations typified by (2) asomni-equations, i.e. equations valid for an arbitrary well-behaved functionΨ ( ). The omni-transform owes its flexibility to the arbitrariness ofΨ ( ) and its ease of handling to its simplicity when applied to mixtures and sums of random variables. From (2) we obtain the moments ofw by puttingΨ (w)=wk, the Laplace transform ofw by puttingΨ(w)=e−sw, and the convolution equation (2a) for the distribution ofw by puttingΨ(w)=1 ifw≥t andΨ(w)=0 otherwise: $$\Pr (w \leqslant t) = (1 - \rho ) + \rho \Pr (w + r \leqslant t),$$ (2a) a result equivalent to the Takacs integro-differential equation. Using repeatedly the so-called shift property of omni-equations, (2a) can be solved by representing the distribution ofw as an infinite series of convolutions: $$\Pr (w \leqslant t) = (1 - \rho ) + (1 - \rho )\rho \Pr (r_1 \leqslant t) + (1 - \rho )\rho ^2 \Pr (r_1 + r_2 \leqslant t) + + ,$$ (3) where theri are a set of independent random variables, each distributed liker. Equation (3) is equivalent to a theorem by Benes. An analogy between the process “waiting time inM/G/1” and the process “toss a coin till heads shows up” where the tossing time is a random variable is also pointed out. The omni-calculus also sheds some light on the modelG/G/1. In forthcoming publications, we will apply the omni-calculus to the process “number in queue” inM/G/1, to the analysis of the busy period inM/G/1, and to some modifiedM/G/1 models, e.g. a vacationing server. In these publications too, the omni-method lifts the “Laplace veil” from much of the physical reality underlying the models considered.